Cubic critical, narrow range

Percentage Accurate: 54.8% → 92.1%

Time: 15.8s

Alternatives: 11

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (if (<= b 0.006)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_0 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_0))))
      (* a 3.0))
     (+
      (* -1.0546875 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0)))
      (+
       (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
       (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.006) {
		tmp = ((pow(-b, 2.0) + (t_0 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-1.0546875 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0))) + ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (c * 3.0d0)
    if (b <= 0.006d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
    else
        tmp = ((-1.0546875d0) * (((a ** 3.0d0) * (c ** 4.0d0)) / (b ** 7.0d0))) + (((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.006) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-1.0546875 * ((Math.pow(a, 3.0) * Math.pow(c, 4.0)) / Math.pow(b, 7.0))) + ((-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	tmp = 0
	if b <= 0.006:
		tmp = ((math.pow(-b, 2.0) + (t_0 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)
	else:
		tmp = (-1.0546875 * ((math.pow(a, 3.0) * math.pow(c, 4.0)) / math.pow(b, 7.0))) + ((-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	tmp = 0.0
	if (b <= 0.006)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-1.0546875 * Float64(Float64((a ^ 3.0) * (c ^ 4.0)) / (b ^ 7.0))) + Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = 0.0;
	if (b <= 0.006)
		tmp = (((-b ^ 2.0) + (t_0 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
	else
		tmp = (-1.0546875 * (((a ^ 3.0) * (c ^ 4.0)) / (b ^ 7.0))) + ((-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.006], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0060000000000000001

   1. Initial program 94.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 94.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. flip-+ 93.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}{3 \cdot a} \]

      2. pow2 93.9%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      3. add-sqr-sqrt 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      4. pow2 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      5. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      6. pow2 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      7. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   6. Applied egg-rr 94.6%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   if 0.0060000000000000001 < b

   1. Initial program 53.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   5. Taylor expanded in b around inf 91.9%

      \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + \left(-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{3 \cdot a} \]

   7. Simplified 91.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, \mathsf{fma}\left(-0.5, \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7}}, -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in a around 0 92.3%

      \[\leadsto \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(a \cdot \left(c \cdot 3\right) - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 2: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (if (<= b 0.006)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_0 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_0))))
      (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.006) {
		tmp = ((pow(-b, 2.0) + (t_0 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (c * 3.0d0)
    if (b <= 0.006d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((a * c) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.006) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((a * c), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	tmp = 0
	if b <= 0.006:
		tmp = ((math.pow(-b, 2.0) + (t_0 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((a * c), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	tmp = 0.0
	if (b <= 0.006)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = 0.0;
	if (b <= 0.006)
		tmp = (((-b ^ 2.0) + (t_0 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((a * c) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.006], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0060000000000000001

   1. Initial program 94.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 94.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. flip-+ 93.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}{3 \cdot a} \]

      2. pow2 93.9%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      3. add-sqr-sqrt 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      4. pow2 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      5. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      6. pow2 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      7. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   6. Applied egg-rr 94.6%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   if 0.0060000000000000001 < b

   1. Initial program 53.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 92.3%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   3. Taylor expanded in c around 0 92.3%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-out 92.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 92.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. *-commutative 92.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. times-frac 92.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right) \]

   5. Simplified 92.3%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(a \cdot \left(c \cdot 3\right) - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternative 3: 85.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0003:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0003)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_0 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_0))))
      (* a 3.0))
     (/ 1.0 (+ (* (/ b c) -2.0) (* 1.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0003) {
		tmp = ((pow(-b, 2.0) + (t_0 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (c * 3.0d0)
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.0003d0)) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (((b / c) * (-2.0d0)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0003) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0003:
		tmp = ((math.pow(-b, 2.0) + (t_0 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)
	else:
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0003)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b / c) * -2.0) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0003)
		tmp = (((-b ^ 2.0) + (t_0 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
	else
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0003], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.99999999999999974e-4

   1. Initial program 76.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 76.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 76.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. flip-+ 76.6%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}{3 \cdot a} \]

      2. pow2 76.6%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      3. add-sqr-sqrt 77.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      4. pow2 77.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      5. associate-*l* 77.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      6. pow2 77.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      7. associate-*l* 77.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   6. Applied egg-rr 77.9%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   if -2.99999999999999974e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 41.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 90.9%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. clear-num 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 90.9%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 90.9%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-def 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 90.9%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}\right)}^{-1} \]

   4. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}} \]

      2. associate-/r/ 90.8%

         \[\leadsto \frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right)}} \]

   6. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(\frac{a}{b} \cdot c\right)\right)}}} \]

   7. Taylor expanded in a around 0 91.5%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0003:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(a \cdot \left(c \cdot 3\right) - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 4: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0003:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0003)
   (pow
    (/ 1.0 (cbrt (/ (* a 3.0) (- (sqrt (fma b b (* a (* c -3.0)))) b))))
    3.0)
   (/ 1.0 (+ (* (/ b c) -2.0) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0003) {
		tmp = pow((1.0 / cbrt(((a * 3.0) / (sqrt(fma(b, b, (a * (c * -3.0)))) - b)))), 3.0);
	} else {
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0003)
		tmp = Float64(1.0 / cbrt(Float64(Float64(a * 3.0) / Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b)))) ^ 3.0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b / c) * -2.0) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0003], N[Power[N[(1.0 / N[Power[N[(N[(a * 3.0), $MachinePrecision] / N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(1.0 / N[(N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.99999999999999974e-4

   1. Initial program 76.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 76.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 76.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 76.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{3 \cdot a}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{3 \cdot a}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{3 \cdot a}}} \]

      2. pow3 76.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{3 \cdot a}}\right)}^{3}} \]

      3. neg-mul-1 76.6%

         \[\leadsto {\left(\sqrt[3]{\frac{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{3 \cdot a}}\right)}^{3} \]

      4. fma-def 76.6%

         \[\leadsto {\left(\sqrt[3]{\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a}}\right)}^{3} \]

      5. pow2 76.6%

         \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}\right)}{3 \cdot a}}\right)}^{3} \]

      6. associate-*l* 76.7%

         \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}\right)}{3 \cdot a}}\right)}^{3} \]

      7. *-commutative 76.7%

         \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{\color{blue}{a \cdot 3}}}\right)}^{3} \]

   6. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a \cdot 3}}\right)}^{3}} \]
   7. Step-by-step derivation

      1. clear-num 76.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}}}\right)}^{3} \]

      2. cbrt-div 76.7%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}}\right)}}^{3} \]

      3. metadata-eval 76.7%

         \[\leadsto {\left(\frac{\color{blue}{1}}{\sqrt[3]{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}}\right)}^{3} \]

   8. Applied egg-rr 76.7%

      \[\leadsto {\color{blue}{\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}}\right)}}^{3} \]
   9. Step-by-step derivation

      1. fma-def 76.7%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\color{blue}{-1 \cdot b + \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}}}\right)}^{3} \]

      2. +-commutative 76.7%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\color{blue}{\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} + -1 \cdot b}}}}\right)}^{3} \]

      3. mul-1-neg 76.7%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} + \color{blue}{\left(-b\right)}}}}\right)}^{3} \]

      4. unsub-neg 76.7%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\color{blue}{\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} - b}}}}\right)}^{3} \]

      5. unpow2 76.7%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\color{blue}{b \cdot b} - a \cdot \left(c \cdot 3\right)} - b}}}\right)}^{3} \]

      6. fma-neg 77.0%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -a \cdot \left(c \cdot 3\right)\right)}} - b}}}\right)}^{3} \]

      7. associate-*r* 76.9%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 3}\right)} - b}}}\right)}^{3} \]

      8. distribute-rgt-neg-in 76.9%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-3\right)}\right)} - b}}}\right)}^{3} \]

      9. metadata-eval 76.9%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-3}\right)} - b}}}\right)}^{3} \]

      10. associate-*r* 77.0%

          \[\leadsto {\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)} - b}}}\right)}^{3} \]

   10. Simplified 77.0%

       \[\leadsto {\color{blue}{\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}}}\right)}}^{3} \]

   if -2.99999999999999974e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 41.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 90.9%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. clear-num 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 90.9%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 90.9%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-def 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 90.9%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}\right)}^{-1} \]

   4. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}} \]

      2. associate-/r/ 90.8%

         \[\leadsto \frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right)}} \]

   6. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(\frac{a}{b} \cdot c\right)\right)}}} \]

   7. Taylor expanded in a around 0 91.5%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0003:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 5: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \mathbf{if}\;b \leq 0.0055:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (if (<= b 0.0055)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_0 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_0))))
      (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.0055) {
		tmp = ((pow(-b, 2.0) + (t_0 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (c * 3.0d0)
    if (b <= 0.0055d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.0055) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	tmp = 0
	if b <= 0.0055:
		tmp = ((math.pow(-b, 2.0) + (t_0 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	tmp = 0.0
	if (b <= 0.0055)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = 0.0;
	if (b <= 0.0055)
		tmp = (((-b ^ 2.0) + (t_0 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0055], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0054999999999999997

   1. Initial program 94.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 94.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. flip-+ 93.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}{3 \cdot a} \]

      2. pow2 93.9%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      3. add-sqr-sqrt 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      4. pow2 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      5. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      6. pow2 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      7. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   6. Applied egg-rr 94.6%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   if 0.0054999999999999997 < b

   1. Initial program 53.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 89.4%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0055:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(a \cdot \left(c \cdot 3\right) - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\left(-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right) + -1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}}{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (if (<= b 0.006)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_0 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_0))))
      (* a 3.0))
     (/
      1.0
      (/
       a
       (/
        (+
         (+
          (* -1.5 (/ a (/ b c)))
          (* -1.125 (/ (pow (* a c) 2.0) (pow b 3.0))))
         (* -1.6875 (/ (pow (* a c) 3.0) (pow b 5.0))))
        3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.006) {
		tmp = ((pow(-b, 2.0) + (t_0 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = 1.0 / (a / ((((-1.5 * (a / (b / c))) + (-1.125 * (pow((a * c), 2.0) / pow(b, 3.0)))) + (-1.6875 * (pow((a * c), 3.0) / pow(b, 5.0)))) / 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (c * 3.0d0)
    if (b <= 0.006d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (a / (((((-1.5d0) * (a / (b / c))) + ((-1.125d0) * (((a * c) ** 2.0d0) / (b ** 3.0d0)))) + ((-1.6875d0) * (((a * c) ** 3.0d0) / (b ** 5.0d0)))) / 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double tmp;
	if (b <= 0.006) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = 1.0 / (a / ((((-1.5 * (a / (b / c))) + (-1.125 * (Math.pow((a * c), 2.0) / Math.pow(b, 3.0)))) + (-1.6875 * (Math.pow((a * c), 3.0) / Math.pow(b, 5.0)))) / 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	tmp = 0
	if b <= 0.006:
		tmp = ((math.pow(-b, 2.0) + (t_0 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)
	else:
		tmp = 1.0 / (a / ((((-1.5 * (a / (b / c))) + (-1.125 * (math.pow((a * c), 2.0) / math.pow(b, 3.0)))) + (-1.6875 * (math.pow((a * c), 3.0) / math.pow(b, 5.0)))) / 3.0))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	tmp = 0.0
	if (b <= 0.006)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(a / Float64(Float64(Float64(Float64(-1.5 * Float64(a / Float64(b / c))) + Float64(-1.125 * Float64((Float64(a * c) ^ 2.0) / (b ^ 3.0)))) + Float64(-1.6875 * Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0)))) / 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = 0.0;
	if (b <= 0.006)
		tmp = (((-b ^ 2.0) + (t_0 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
	else
		tmp = 1.0 / (a / ((((-1.5 * (a / (b / c))) + (-1.125 * (((a * c) ^ 2.0) / (b ^ 3.0)))) + (-1.6875 * (((a * c) ^ 3.0) / (b ^ 5.0)))) / 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.006], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a / N[(N[(N[(N[(-1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.6875 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0060000000000000001

   1. Initial program 94.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 94.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 94.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. flip-+ 93.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}{3 \cdot a} \]

      2. pow2 93.9%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      3. add-sqr-sqrt 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      4. pow2 94.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      5. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      6. pow2 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

      7. associate-*l* 94.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   6. Applied egg-rr 94.6%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}}{3 \cdot a} \]

   if 0.0060000000000000001 < b

   1. Initial program 53.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. add-log-exp 51.2%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}\right)}}{3 \cdot a} \]

      2. neg-mul-1 51.2%

         \[\leadsto \frac{\log \left(e^{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}\right)}{3 \cdot a} \]

      3. fma-def 51.2%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}\right)}{3 \cdot a} \]

      4. pow2 51.2%

         \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}\right)}\right)}{3 \cdot a} \]

      5. associate-*l* 51.2%

         \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}\right)}\right)}{3 \cdot a} \]

   6. Applied egg-rr 51.2%

      \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. clear-num 51.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}}} \]

      2. inv-pow 51.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)}^{-1}} \]

      3. *-commutative 51.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)}^{-1} \]

      4. rem-log-exp 53.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}\right)}^{-1} \]

   8. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 53.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}} \]

      2. associate-/l* 53.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{3}}}} \]

      3. *-commutative 53.2%

         \[\leadsto \frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \color{blue}{\left(3 \cdot c\right)}}\right)}{3}}} \]

      4. associate-*r* 53.3%

         \[\leadsto \frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 3\right) \cdot c}}\right)}{3}}} \]

   10. Simplified 53.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{3}}}} \]

   11. Taylor expanded in a around 0 45.6%

       \[\leadsto \frac{1}{\frac{a}{\frac{\color{blue}{b + \left(-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + \left(-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1 \cdot b\right)\right)\right)}}{3}}} \]
   12. Step-by-step derivation

       1. +-commutative 45.6%

          \[\leadsto \frac{1}{\frac{a}{\frac{b + \color{blue}{\left(\left(-1.5 \cdot \frac{a \cdot c}{b} + \left(-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1 \cdot b\right)\right) + -1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}\right)}}{3}}} \]

       2. associate-+r+ 46.3%

          \[\leadsto \frac{1}{\frac{a}{\frac{\color{blue}{\left(b + \left(-1.5 \cdot \frac{a \cdot c}{b} + \left(-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1 \cdot b\right)\right)\right) + -1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}}}{3}}} \]

   13. Simplified 89.0%

       \[\leadsto \frac{1}{\frac{a}{\frac{\color{blue}{\left(\left(0 + -1.5 \cdot \frac{a}{\frac{b}{c}}\right) + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right) + -1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}}}{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(a \cdot \left(c \cdot 3\right) - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\left(-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right) + -1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}}{3}}}\\ \end{array} \]

Alternative 7: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0003:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0003)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) a))
   (/ 1.0 (+ (* (/ b c) -2.0) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0003) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (a * (c * -3.0)))) - b) / a);
	} else {
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0003)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / a));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b / c) * -2.0) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0003], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.99999999999999974e-4

   1. Initial program 76.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0 76.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

   4. Simplified 76.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. add-log-exp 70.0%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}\right)}}{3 \cdot a} \]

      2. neg-mul-1 70.0%

         \[\leadsto \frac{\log \left(e^{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}\right)}{3 \cdot a} \]

      3. fma-def 70.0%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}\right)}{3 \cdot a} \]

      4. pow2 70.0%

         \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}\right)}\right)}{3 \cdot a} \]

      5. associate-*l* 70.0%

         \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}\right)}\right)}{3 \cdot a} \]

   6. Applied egg-rr 70.0%

      \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. clear-num 70.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}}} \]

      2. inv-pow 70.0%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)}^{-1}} \]

      3. *-commutative 70.0%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)}^{-1} \]

      4. rem-log-exp 76.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}\right)}^{-1} \]

   8. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 76.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}} \]

      2. associate-/l* 76.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{3}}}} \]

      3. *-commutative 76.6%

         \[\leadsto \frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \color{blue}{\left(3 \cdot c\right)}}\right)}{3}}} \]

      4. associate-*r* 76.7%

         \[\leadsto \frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 3\right) \cdot c}}\right)}{3}}} \]

   10. Simplified 76.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{3}}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 43.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{3}}}\right)\right)} \]

       2. expm1-udef 43.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{3}}}\right)} - 1} \]

       3. associate-/r/ 43.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{a} \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{3}}\right)} - 1 \]

       4. div-inv 43.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \color{blue}{\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right) \cdot \frac{1}{3}\right)}\right)} - 1 \]

       5. associate-*l* 43.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{a \cdot \left(3 \cdot c\right)}}\right) \cdot \frac{1}{3}\right)\right)} - 1 \]

       6. *-commutative 43.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \color{blue}{\left(c \cdot 3\right)}}\right) \cdot \frac{1}{3}\right)\right)} - 1 \]

       7. metadata-eval 43.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right) \cdot \color{blue}{0.3333333333333333}\right)\right)} - 1 \]

   12. Applied egg-rr 43.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right) \cdot 0.3333333333333333\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 43.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a} \cdot \left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right) \cdot 0.3333333333333333\right)\right)\right)} \]

       2. expm1-log1p 76.7%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right) \cdot 0.3333333333333333\right)} \]

       3. associate-*l/ 76.7%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right) \cdot 0.3333333333333333\right)}{a}} \]

       4. *-lft-identity 76.7%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right) \cdot 0.3333333333333333}}{a} \]

       5. *-commutative 76.7%

          \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}{a} \]

       6. *-lft-identity 76.7%

          \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{\color{blue}{1 \cdot a}} \]

       7. times-frac 76.6%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{1} \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a}} \]

       8. metadata-eval 76.6%

          \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a} \]

   14. Simplified 76.9%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a}} \]

   if -2.99999999999999974e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 41.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 90.9%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. clear-num 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 90.9%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 90.9%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-def 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 90.9%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}\right)}^{-1} \]

   4. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}} \]

      2. associate-/r/ 90.8%

         \[\leadsto \frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right)}} \]

   6. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(\frac{a}{b} \cdot c\right)\right)}}} \]

   7. Taylor expanded in a around 0 91.5%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0003:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 8: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.0003:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0003) t_0 (/ 1.0 (+ (* (/ b c) -2.0) (* 1.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0003) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.0003d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((b / c) * (-2.0d0)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0003) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.0003:
		tmp = t_0
	else:
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.0003)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b / c) * -2.0) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.0003)
		tmp = t_0;
	else
		tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0003], t$95$0, N[(1.0 / N[(N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.99999999999999974e-4

   1. Initial program 76.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   if -2.99999999999999974e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 41.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 90.9%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. clear-num 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 90.9%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 90.9%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-def 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 90.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 90.9%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}\right)}^{-1} \]

   4. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 90.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}} \]

      2. associate-/r/ 90.8%

         \[\leadsto \frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right)}} \]

   6. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(\frac{a}{b} \cdot c\right)\right)}}} \]

   7. Taylor expanded in a around 0 91.5%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0003:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 9: 82.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* (/ b c) -2.0) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b / c) * (-2.0d0)) + (1.5d0 * (a / b)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
}

Python

def code(a, b, c):
	return 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(b / c) * -2.0) + Float64(1.5 * Float64(a / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((b / c) * -2.0) + (1.5 * (a / b)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in b around inf 80.6%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. clear-num 80.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

   2. inv-pow 80.6%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

   3. *-commutative 80.6%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

   4. +-commutative 80.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

   5. fma-def 80.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

   6. div-inv 80.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   7. pow-prod-down 80.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   8. pow-flip 80.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   9. metadata-eval 80.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   10. associate-/l* 80.7%

       \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}\right)}^{-1} \]

4. Applied egg-rr 80.7%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}\right)}^{-1}} \]
5. Step-by-step derivation

   1. unpow-1 80.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}} \]

   2. associate-/r/ 80.6%

      \[\leadsto \frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right)}} \]

6. Simplified 80.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(\frac{a}{b} \cdot c\right)\right)}}} \]

7. Taylor expanded in a around 0 81.5%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]

8. Final simplification 81.5%

   \[\leadsto \frac{1}{\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}} \]

Alternative 10: 64.9% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in b around inf 64.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]

3. Final simplification 64.4%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]

Alternative 11: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in a around 0 55.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. *-commutative 55.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]

4. Simplified 55.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. add-log-exp 52.3%

      \[\leadsto \frac{\color{blue}{\log \left(e^{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}\right)}}{3 \cdot a} \]

   2. neg-mul-1 52.3%

      \[\leadsto \frac{\log \left(e^{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}\right)}{3 \cdot a} \]

   3. fma-def 52.3%

      \[\leadsto \frac{\log \left(e^{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}\right)}{3 \cdot a} \]

   4. pow2 52.3%

      \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3}\right)}\right)}{3 \cdot a} \]

   5. associate-*l* 52.4%

      \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 3\right)}}\right)}\right)}{3 \cdot a} \]

6. Applied egg-rr 52.4%

   \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}\right)}}{3 \cdot a} \]

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023325 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))